A well-known problem in metrology involves the interferometric measurement of distances and of changes in distance for surfaces that may have discontinuities larger than the source wavelength. Calculation of a round-trip optical path difference L in interferometers is carried out on the basis of the measured phase -.pi.&lt;.phi.&lt;.pi. and the optical wavelength .lambda.: ##EQU1## Since the phase is determined from an inverse trigonometric function, the same measured value of .phi. will be repeated at path-length intervals R.sub.n equal to EQU R.sub.n =n.lambda. (Equation 2)
where n is an integer that cannot be determined from the phase measurement alone. In the absence of some other information relating to the optical path length, the interferometer can only make unambiguous single-point measurements over a range limited to the optical wavelength.
To increase the unambiguous range, a second wavelength .lambda..sub.2 with an associated phase .phi..sub.2 and integer fringe order n.sub.2 may be introduced. Multiple-color interferometry has been used for over a century to facilitate the identification of fringe orders. Traditionally, the analysis proceeds by some variation of the method of excess fractions, also known as the method of exact fractions which, according to the text Principles of Optics by M. Born and E. Wolf (Pergamon Press, 1987), was first used in four-color interferometry by J. R. Benoit in 1898. The method of excess fractions consists of determining mutually consistent values for the integer fringe orders n and n.sub.2, given the measured fractional fringe orders .phi./2.pi. and .phi..sub.2 /2.pi.. In its simplest form, the computational procedure for this method can be reduced to calculating a large number of possible distances for the given fractional fringe orders, and then observing which values are in closest agreement.
It has become common practice in the art to make the data processing in two-color interferometry more rapid and intuitive by defining a synthetic wavelength .LAMBDA., defined by the spatial beat period for a two-color interference pattern. The corresponding synthetic phase is ##EQU2## constrained by -.pi.&lt;.PHI..ltoreq..pi.. Using this concept and assuming a perfectly compensated interferometer, a distance L' can be obtained from the relationship ##EQU3## where N is the integer synthetic fringe order and the synthetic wavelength .LAMBDA. is defined by ##EQU4## If N=0, then an estimate n' of the optical wavelength fringe order can be made by substituting Eq. (4) into Eq. (1) and rearranging to obtain ##EQU5## The final distance measurement is then ##EQU6## where the function Int{} returns the nearest integer to its argument. The unambiguous measurement range has thus now been extended to the synthetic wavelength .LAMBDA., which may be very much larger than .lambda..
The use of synthetic wavelengths has been widely accepted in many different forms of interferometry. In the two-wavelength holographic method described by K. Haines and B. P. Hildebrand in Contour Generation By Wavefront Reconstruction, 19 Physics Letters 10-11 (1965), the synthetic wavelength corresponds to the contour intervals of constructive interference in the reconstructed holographic image. Similar techniques involving synthetic wavelengths have been described by J. C. Wyant in Testing Aspherics Using Two-Wavelength Holography, 10 Applied Optics 2113-18 (1971). A computational approach to the method of exact fractions based on synthetic wavelengths is described in an article by C. R. Tilford entitled Analytical Procedure For Determining Lengths From Fractional Fringes, 16 Applied Optics 1857-60 (1977). U.S. Pat. No. 4,355,899 of T. A. Nussmeier, entitled Interferometric Distance Measurement Method, discloses the general concept of using synthetic wavelength information to remove the phase ambiguities in interferometry. This principle has been applied to full-aperture phase-modulation interferometry by Y. Cheng and J. C. Wyant, as described in Two-Wavelength Phase Shying Interferometry, 23 Applied Optics 4539-43 (1984). Virtually all modern embodiments of multiple color interferometers employ an analysis based on synthetic wavelengths as is manifest, for example, from the review article Absolute Distance Interferometry by N. A. Massie and H. John Caulfield, 816 Proceedings of the Society of Photooptical Engineers 149-57 (1987).
The only limitation of the synthetic-wavelength method, apart from practical difficulties in construction of appropriate instrumentation, is that the same synthetic phase .PHI. will repeat itself at distances EQU R.sub.n =N.LAMBDA.. (Equation 8)
Thus, the unambiguous range interval for this method is defined by .vertline.L.vertline.&lt;.LAMBDA./2. It is generally accepted in the art that the only way in which to extend this range is to either increase the synthetic wavelength .LAMBDA. or incorporate additional optical, electrical or mechanical means for removing the synthetic-wavelength phase ambiguity. This limitation, which is evident in the aforementioned articles and widely known to those skilled in the art, restricts the choice of source and detection methods available for the implementation of two-color interferometry. For example the two-color source, which may be formed by two different kinds of inexpensive lasers, light-emitting diodes or interference filters, can become more costly or difficult to implement if a significantly different wavelength separation is required in order to increase the synthetic wavelength. In addition, some interferometers use prisms, gratings or interference filters to separate the two colors before detection, and this separation is rendered more difficult when the two colors are selected as very close in wavelength to one another in order to generate a large synthetic wavelength. These problems apply to all interferometers employing two colors for the purpose of measuring distances unambiguously over ranges larger than an optical wavelength.